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PreprintCC BY 4.0arXiv

On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$

  • Mohamed Mahmoud Chems-Eddin
  • Ikrame Daqaq
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Abstract

We investigate the stability of the $2$-rank in the cyclotomic ℤ2-extension of real biquadratic number fields. We determine several families of real biquadratic fields F for which the ranks of the $2$-class group A(F) and the associated $2$-Iwasawa module A∞(F) are both equal to $3$. More precisely, we provide a complete classification of all fields F=ℚ(√q,√d), where q≡ 3pmod4 is a prime number and d is an odd positive square-free integer, and all fields F=ℚ(√(2q),√d), where q≡ 3pmod4 is a prime number and d≡ 1pmod4 is a positive square-free integer, for which $ \operatorname{rank}(A(F))=\operatorname{rank}(A_\infty(F))=3.$

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Cite

BibTeXRIS
BibTeX
@misc{chemseddin2026real,
  title = {{On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$}},
  author = {Mohamed Mahmoud Chems-Eddin and Ikrame Daqaq},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34400},
  eprint = {2610.34400},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34400},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$
AU  - Mohamed Mahmoud Chems-Eddin
AU  - Ikrame Daqaq
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34400
UR  - https://arxiv.org/abs/2610.34400
AB  - We investigate the stability of the $2$-rank in the cyclotomic ℤ_2-extension of real biquadratic number fields. We determine several families of real biquadratic fields F for which the ranks of the $2$-class group A(F) and the associated $2$-Iwasawa module A_∞(F) are both equal to $3$. More precisely, we provide a complete classification of all fields F=ℚ(√q,√d), where q≡ 3pmod4 is a prime number and d is an odd positive square-free integer, and all fields F=ℚ(√(2q),√d), where q≡ 3pmod4 is a prime number and d≡ 1pmod4 is a positive square-free integer, for which $ \operatorname{rank}(A(F))=\operatorname{rank}(A_\infty(F))=3.$
ER  -